A variational data assimilation approach for sparse velocity reference data in coarse RANS simulations through a corrective forcing term

Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three‐dimensional and four‐dimensional variational data assimilation (3D‐Var and 4D‐Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

Enhancing CFD-LES air pollution prediction accuracy using data assimilation

Mean wind flow reconstruction of a high-rise building based on variational data assimilation using sparse pressure measurements

In different studies, land surface temperature (LST) observations have been assimilated into the variational data assimilation (VDA) approaches to estimate turbulent heat fluxes. The VDA methods yield accurate turbulent heat fluxes, but they need an adjoint model, which is difficult to derive and code. They also cannot directly calculate the uncertainty of their estimates. To overcome the abovementioned drawbacks, this study assimilates LST data from Geostationary Operational Environmental Satellite into the ensemble Kalman smoother (EnKS) data assimilation system to estimate turbulent heat fluxes. EnKS does not need to derive the adjoint term and directly generates statistical information on the accuracy of its predictions. It uses the heat diffusion equation to simulate LST. EnKS with the state augmentation approach finds the optimal values for the unknown parameters (i.e., evaporative fraction and neutral bulk heat transfer coefficient, CHN) by minimizing the misfit between LST observations from Geostationary Operational Environmental Satellite and LST estimations from the heat diffusion equation. The augmented EnKS scheme is tested over six Ameriflux sites with a wide range of hydrological and vegetative conditions. The results show that EnKS can predict not only the model parameters and turbulent heat fluxes but also their uncertainties over a variety of land surface conditions. Compared to the variational method, EnKS yields suboptimal turbulent heat fluxes. However, suboptimality of EnKS is small, and its results are comparable to those of the VDA method. Overall, EnKS is a feasible and reliable method for estimation of turbulent heat fluxes.

Within the EURANOS project data assimilation (DA) approaches have been successfully applied in two areas to improve the predictive power of simulation models used in the RODOS and ARGOS decision support systems. For the areas of atmospheric dispersion modelling and of modelling the fate of radio-nuclides in urban areas the results of demonstration exercises are presented here. With the data assimilation module of the RIMPUFF dispersion code, predictions of the gamma dose rate are corrected with simulated readings of fixed detector stations. Using the DA capabilities of the IAMM package for mapping the radioactive contamination in inhabited areas, predictions of a large scale deposition model have been combined with hypothetical measurements on a local scale. In both examples the accuracy of the model predictions has been improved and the uncertainties have been reduced.

Pertaining the application of the 4DVar and 4DEnVar WRFDA techniques to simulate tropical cyclones in the Bay of Bengal

Data assimilation (DA) has shown to be a powerful method to improve hydronumeric models by integration of observations. However, in systems which are strongly governed by boundary conditions, the benefits of DA are typically limited to improvements of hindcast and quickly fade in forecasts where no observations are available. Where sufficient historical observations are available, machine learning (ML) models can be an attractive alternative for providing accurate forecasts of hydrodynamic conditions.With the goal to improve the forecast, we propose a hybrid approach combining a physics-based numerical model with a machine learning approach via data assimilation. The ML model delivers accurate forecasts in a few points where historical observations are available. These results are then treated as synthetic observations by data assimilation which transfers the improvements to surrounding positions and other model variables.This approach is illustrated with a case-study from the Elbe Estuary. Here, an operational two-dimensional hydronumeric model, strongly driven by boundary conditions, is used to compute the hydraulic conditions in the estuary with the main purpose of ensuring safe nautical navigation. The existing model is extended with an Ensemble Kalman Filter data assimilation approach where an ensemble is created by stochastic perturbation of model forcings.During hindcast, data from two stations of the estuary are assimilated to improve numerical model results and initial conditions for the forecast. To maximize forecasting skill, a Long-short-term-memory machine learning approach is used to provide synthetic observations at assimilation stations during the forecast.Results from the hybrid model compared to a baseline model, without assimilation, at independent validation stations show that the hybrid model can reduce forecast errors by 40% for water levels and prolongs the positive influence of data assimilation significantly.

Quantifying continental-scale river discharge is essential to understanding the terrestrial water cycle but is susceptible to errors caused by a lack of observations and the limitations of hydrodynamic modeling. Data assimilation (DA) methods are increasingly used to estimate river discharge in combination with emerging river-related remote sensing products (e.g., water surface elevation [WSE], water surface slope, river width, and flood extent). However, directly comparing simulated WSE to satellite altimetry data remains challenging (e.g., because of large biases between simulations and observations or uncertainties in parameters), and large errors can be introduced when satellite observations are assimilated into hydrodynamic models. In this study we performed direct, anomaly, and normalized value assimilation experiments to investigate the capacity of DA to improve river discharge within the current limitations of hydrodynamic modeling. We performed hydrological DA using a physically-based empirical localization method applied to the Amazon Basin. We used satellite altimetry data from ENVISAT, Jason 1, and Jason 2. Direct DA was the baseline assimilation method and was subject to errors due to biases in the simulated WSE. To overcome these errors, we used anomaly DA as an alternative to direct DA. We found that the modeled and observed WSE distributions differed considerably (e.g., differences in amplitude, seasonal flow variation, and a skewed distribution due to limitations of the hydrodynamic models). Therefore, normalized value DA was performed to improve discharge estimation. River discharge estimates were improved at 24 %, 38 %, and 62 % of stream gauges in the direct, anomaly, and normalized value assimilations relative to simulations without DA. Normalized value assimilation performed best for estimating river discharge given the current limitations of hydrodynamic models. Most gauges within the river reaches covered by satellite observations accurately estimated river discharge, with Nash-Sutcliffe efficiency (NSE) > 0.6. The amplitudes of WSE variation were improved in the normalized DA experiment. Furthermore, in the Amazon Basin, normalized assimilation (median NSE = 0.50) improved river discharge estimation compared to open-loop simulation with the global hydrodynamic model (median NSE = 0.42). River discharge estimation using direct DA methods was improved by 7 % with calibration of river bathymetry based on NSE. The direct DA approach outperformed the other DA approaches when runoff was considerably biased, but anomaly DA performed best when the river bathymetry was erroneous. The uncertainties in hydrodynamic modeling (e.g., river bottom elevation, river width, simplified floodplain dynamics, and the rectangular cross-section assumption) should be improved to fully realize the advantages of river discharge DA through the assimilation of satellite altimetry. This study contributes to the development of a global river discharge reanalysis product that is consistent spatially and temporally.

Reconstructing urban wind flows for urban air mobility using reduced-order data assimilation

Forecast Sensitivity-based Observation Impact (FSOI) in an analysis–forecast system of the California Current Circulation

This study investigated the feasibility of partitioning the available energy between sensible (H) and latent (LE) heat fluxes via variational assimilation of reference-level air temperature and specific humidity. For this purpose, sequences of reference-level air temperature and specific humidity were assimilated into an atmospheric boundary layer model (ABL) within a variational data assimilation (VDA) framework to estimate H and LE. The VDA approach was tested at six sites (namely, Arou, Audubon, Bondville, Brookings, Desert, and Willow Creek) with contrasting climatic and vegetative conditions. The unknowns of the VDA system were the neutral bulk heat transfer coefficient (CHN) and evaporative fraction (EF). EF estimates were found to agree well with observations in terms of magnitude and day-to-day fluctuations in wet/densely vegetated sites but degraded in dry/sparsely vegetated sites. Similarly, in wet/densely vegetated sites, the variations in the CHN estimates were found to be consistent with those of the leaf area index (LAI) while this consistency deteriorated in dry/sparely vegetated sites. The root mean square errors (RMSEs) of daily H and LE estimates at the Arou site (wet) were 25.43 (Wm−2) and 55.81 (Wm−2), which are respectively 57.6% and 45.4% smaller than those of 60.00 (Wm−2) and 102.21 (Wm−2) at the Desert site (dry). Overall, the results show that the VDA system performs well at wet/densely vegetated sites (e.g., Arou and Willow Creek), but its performance degrades at dry/slightly vegetated sites (e.g., Desert and Audubon). These outcomes show that the sequences of reference-level air temperature and specific humidity have more information on the partitioning of available energy between the sensible and latent heat fluxes in wet/densely vegetated sites than dry/slightly vegetated sites.

Ensemble simulations have been performed with a climate model constrained to follow temperature histories obtained from a recent compilation of 56 well‐calibrated surface temperature proxy records, using a new data assimilation technique. First, we demonstrate that the data assimilation technique provides a faithful representation in the Northern Hemisphere of the signal recorded by the climate proxies at both the regional and gridbox scales. Second, by varying the external forcing, the parameters of the data assimilation method, and the parameters controlling the equilibrium climate sensitivity of the climate model, we demonstrate that the uncertainty in model results is much lower in simulations using data assimilation than in those without it. This observation implies that the data assimilation, using a set of 56 proxies, is providing an efficient and robust constraint on the simulated climate variability over the past centuries. At the hemispheric and continental scales, the model reconstructions using data assimilation are in good agreement with both the instrumental record of the past 150 years and reconstructions of climate in past centuries derived from the application of traditional statistical approaches to networks of proxy data. This increases the confidence in both the data assimilation and traditional statistical approaches. Our data assimilation method, however, is unable to provide a reliable reconstruction over the North Atlantic Ocean, which we attribute to the paucity of proxy data in that region.

The 2010 Eyjafjallajökull volcano eruption had serious consequences to civil aviation. This has initiated a lot of research on volcanic ash forecasting in recent years. For forecasting the volcanic ash transport after eruption onset, a volcanic ash transport and diffusion model (VATDM) needs to be run with Eruption Source Parameters (ESPs) such as plume height and mass eruption rate as input, and with data assimilation techniques to continuously improve the forecast. Reliable and accurate ash measurements are crucial for providing successful ash clouds advices. In the first<br/>phase of this research work, simulated aircraft-based volcanic ash measurements, will be assimilated into a transport model to identify the potential benefit of this kind of observations in an assimilation system. The results show that assimilating aircraft-based measurements can improve the state of ash clouds, and can provide an improved forecast. We also show that for an advice on the aeroplane flying level, aircraft-based measurements should preferably be taken at this level. Furthermore<br/>it is shown that in order to make an acceptable advice for aviation decision makers, accurate knowledge about uncertainties of ESPs and measurements is of great importance.<br/><br/>The forecast accuracy of distal volcanic ash clouds is important for providing valid aviation advice during volcanic ash eruptions. However, because the distal part of a volcanic ash plume is far from the volcano, the influence of eruption information on this part becomes rather indirect and uncertain, resulting in inaccurate volcanic ash forecasts in these distal areas. In this thesis, we use real-life aircraft in situ observations, measured in the North-West part of Germany during the 2010 Eyjafjallajökull eruption, in an ensemble-based data assimilation system to investigate the potential improvement on the forecast accuracy with regard to the distal volcanic ash plume. We show that the error of the analyzed volcanic ash state can be significantly reduced by assimilating real-life in situ measurements. After assimilation, it is shown that the model-based aviation advice for Germany, the Netherlands and Luxembourg can be improved. We suggest that with suitable aircrafts measuring once per day across the distal volcanic ash plume, the description and prediction of volcanic ash clouds in these areas can be improved significantly.<br/><br/>Among the data assimilation approaches, the ensemble Kalman filter (EnKF) is a well-known and popular method. A proper covariance localization strategy in the analysis step of EnKF is essential for reducing spurious covariances caused by the finite ensemble size, as shown for this application for assimilation of aircraft in situ measurements. After analyzing the characteristics of the physical forecast error covariances, we present a two-way tracking approach to define the localization matrix<br/>for covariance localization. The result shows that the Two-way-tracking Localized EnKF (TL-EnKF) effectively maintains the correctly specified physical covariances and largely reduces the spurious ones. The computational cost of TL-EnKF is also evaluated and is shown to be advantageous for both serial and parallel implementations. Compared to the commonly used distance-based covariance localization, the two-way tracking approach is shown to be more suitable. In addition, the covariance inflation approach is verified as an additional improvement to TL-EnKF to achieve more accurate results.<br/><br/>A timely prediction requires that the computations of the data assimilation system can be performed quickly (at least than the Wall-clock). We therefore investigate strategies for accelerating the data assimilation algorithm. Based on evaluations of the computational time, the analysis step of the assimilation turns out to be the most expensive part. After a study on the characteristics of the ensemble ash state, we propose a mask-state algorithm which records the sparsity information of the full ensemble state matrix and transforms the full matrix into a relatively small one. This will reduce the computational cost in the analysis step. Experimental results show the mask-state algorithm significantly speeds up the analysis step. Subsequently, the total amount of computing time for volcanic ash data assimilation is reduced to an acceptable level. The mask-state algorithm is generic and thus can be embedded in any ensemble-based data assimilation framework. Moreover, ensemble-based data assimilation with the mask-state algorithm is promising and flexible, because it implements exactly the standard data assimilation without any approximation and it realizes the satisfying performance without any change of the full model.<br/><br/>Infrared satellite measurements of volcanic ash mass loadings are often used as<br/>input observations for the assimilation scheme. However, these satellite-retrieved<br/>data are often two-dimensional (2D), and cannot easily be combined with a three-dimensional (3D) volcanic ash model to improve the volcanic ash state. By integrating available data including ash mass loadings, cloud top heights and thickness information, we propose a satellite observational operator (SOO) that translates satellite-retrieved 2D volcanic ash mass loadings to 3D concentrations at the top layer of the ash cloud. Ensemble-based data assimilation is used to assimilate the extracted measurements of ash concentrations. The results show that satellite data assimilation can force the volcanic ash state to match the satellite observations, and that it improves the forecast of the ash state. Comparison with highly accurate aircraft in situ measurements shows that the effective duration of the improved volcanic ash forecasts is about a half day.

In two recent studies, the authors presented a new data assimilation (DA) method for precipitation observations that does not require Gaussianity or linearity assumptions. The method, called localized ensemble mosaic assimilation (LEMA), initializes the new ensemble forecast by relaxing the background ensemble (prior) toward a single analysis composed of different column states taken from the ensemble members with the lowest error in the precipitation forecast. However, a limitation of the LEMA method is that relaxing the background ensemble toward that analysis severely reduces the spread of the ensemble, thus, limiting its usefulness for cycled DA applications. This study presents a new version of LEMA, called localized ensemble mosaic assimilation sequence (LEMAS), suitable for cycled DA operations. LEMAS constructs an ensemble of analysis mosaics using a small group of members closer to the observations instead of only the closest one. The new ensemble forecast is then initialized by recentering the prior ensemble around the mean of the analysis ensemble while scaling the original background perturbations to match the spread of the analysis mosaics. A series of ideal and real DA experiments are used to evaluate the potential of LEMAS for the assimilation of hourly accumulation observations. A comparison of LEMAS with the local ensemble transform Kalman filter (LETKF) using idealized experiments shows that LEMAS produces similar or slightly better forecast quality than the LETKF in temperature, water vapor, winds, and precipitation. Extending this comparison to real DA experiments assimilating Stage-IV precipitation observations shows that both methods produce precipitation forecasts of comparable quality.

We apply information theory within an ensemble‐based data assimilation approach and define information matrix in ensemble subspace. The information matrix in ensemble subspace employs a flow‐dependent forecast error covariance and it is of relatively small dimensions (equal to the ensemble size). The information matrix in ensemble subspace can be directly linked to the information matrix typically used in non‐ensemble‐based data assimilation methods, such as the Kalman Filter (KF) and the three‐dimensional variational (3D‐Var) methods, which provides a framework for consistent comparisons of information measures between different data assimilation methods.We evaluate information measures, such as degrees of freedom for signal, within the Maximum Likelihood Ensemble Filter (MLEF) data assimilation approach and compare them with those obtained using the KF approach and the 3D‐Var approach. We assimilate model‐simulated observations and use the Goddard Earth Observing System Single Column Model (GEOS‐5 SCM) as a dynamical forecast model.The experimental results demonstrate that the proposed framework is useful for comparing information measures obtained in different data assimilation approaches. These comparisons indicate that using a flow‐dependent forecast error covariance matrix (e.g. as in the KF and the MLEF experiments) is fundamentally important for adequately describing prior knowledge about the true model state when calculating information measures of assimilated observations. We also demonstrate that data assimilation results obtained using the KF and the MLEF approach (when ensemble size is larger than 10 ensemble members) are superior to the results of the 3D‐Var approach. Copyright © 2007 Royal Meteorological Society